Extreme and Singular Behavior in Fundamental Models of Fluid Mechanics

这个数学研究项目将开发和应用理论和计算工具，系统地寻找一些流体力学基本方程的解的极端行为。其目标是从第一性原理中推导出物理上有意义的量的精确预测，并理解潜在的非线性动力学机制。这项工作充分利用了首席研究员、合作者和其他科学家的最新发展，以实施最优控制理论和变分法的思想，以找到实现最大混合、最优输送或其他极端耗散的流动。输运、混合和耗散是流体流动最基本的特征之一，它们对微流体工程、气候科学和天体物理等重要应用具有基础性意义。这里采用的控制和优化技术构成了对这些问题的新的计算辅助分析方法。该项目直接涉及对密歇根大学首席研究员S指导下的研究生和博士后研究人员的高级培训。研究重点是流体力学中偏微分方程解的定性和定量性质，包括纳维尔-斯托克斯方程和平流-扩散方程。研究人员与密歇根大学的研究生和博士后研究人员以及其他地方的研究人员合作，使用现代应用分析和科学计算。这项工作的四个主要部分是：(1)研究对流扩散方程解的定性和定量性质，并根据搅拌流的特征确定混合极限，以了解湍流的混合有效性；(2)利用新的优化方案研究非线性动力系统中的极端时间平均，并研究它们对经典Rayleigh-B？nard对流模型的影响；(3)实施计算最优控制方法，在三维Navier-Stokes方程中寻找最大拟能产生量；以及(4)将分析和计算技术应用于生物、化学和物理的跨学科应用中的其他非线性和/或随机动力学模型。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This mathematical research project will develop and apply theoretical and computational tools to systematically search for extreme behavior in solutions of some of the fundamental equations of fluid mechanics. The goal is to derive precise predictions of physically significant quantities from first principles and understand the underlying nonlinear dynamical mechanisms. The work capitalizes on recent developments of the Principal Investigator, collaborators, and other scientists to implement ideas from optimal control theory and the calculus of variations to find flows achieving maximal mixing, optimal transport, or other extreme dissipation. Transport, mixing, and dissipation are among the most fundamental features of fluid flows and they are of foundational significance for important applications ranging from microfluidics engineering to climate science and astrophysics. The control and optimization techniques adopted here constitute novel computationally aided analysis approaches to these problems. This project directly involves advanced training for graduate students and postdoctoral researchers under the Principal Investigator?s direction at the University of Michigan.The research is centered on studies of qualitative and quantitative properties of solutions of partial differential equations in fluid mechanics including the Navier-Stokes and advection-diffusion equations. The investigator employs modern applied analysis and scientific computation in collaboration with graduate students and postdoctoral researchers at the University of Michigan, and with other researchers elsewhere. The four major components of the work are: (1) to study qualitative and quantitative properties of solutions to advection-diffusion equations and determine limits on mixing in terms of features of the stirring flows with an ultimate goal of understanding mixing effectiveness of turbulence; (2) to utilize new optimization schemes to investigate extreme time averages in nonlinear dynamical systems and investigate their implications for classical models of Rayleigh-B?nard convection; (3) to implement computational optimal control methods to search for maximal enstrophy production in the three-dimensional Navier-Stokes equations; and (4) to apply the analysis and computation techniques to other nonlinear and/or stochastic dynamical models from interdisciplinary applications in biology, chemistry and physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.